For what negative value of k is there exactly one solution to the system of equations \(\begin{align*} y &= 2x^2 + kx + 6 \\ y &= -x + 4? \end{align*}\)
The quadratic x^2-3x+1 can be written in the form (x+b)^2+c, where b and c are constants. What is b+c?
A parabola with equation y=x^2+bx+c passes through the points (-1,-11) and (3,17). What is c?
What is the value of c if \(x\cdot(3x+1)<c\) if and only when \(x\in \left(-\frac{7}{3},2\right)\)?
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\(\begin{align*} y &= 2x^2 + kx + 6 \\ y &= -x + 4? \end{align*}\\~\\ y=2x^2+kx+6\\ -x+4=2x^2+kx+6\\ 0=2x^2+kx+x+6-4\\ 0=2x^2+(k+1)x+2\\ \mbox{If there is to be only one solution, the discriminant must equal 0}\\ \triangle=(k+1)^2-16=0\\ (k+1)^2=16\\ k+1=\pm4\\ \mbox{The only negative solution for k is }\\ k=-5 \)
Here is the graph.
+118658 It is best to ask just one question at a time.
Learn from the answer or ask questions about it.
THEN
move to the next question if you still need help with it.
\(\begin{align*} y &= 2x^2 + kx + 6 \\ y &= -x + 4? \end{align*}\\~\\ y=2x^2+kx+6\\ -x+4=2x^2+kx+6\\ 0=2x^2+kx+x+6-4\\ 0=2x^2+(k+1)x+2\\ \mbox{If there is to be only one solution, the discriminant must equal 0}\\ \triangle=(k+1)^2-16=0\\ (k+1)^2=16\\ k+1=\pm4\\ \mbox{The only negative solution for k is }\\ k=-5 \)
Here is the graph.
y=x2+bx+c, passes through (-1, -11) and (3, 17). c=?
A. 11=1-b+c
B. 17=9+3b+c
C. c=b-12
D. 17=4b-3->4b=20
b=5
c=-7
x2-3x+1=(x+b)2+c
b+c=?
(x+b)2+c=x2+2bx+b2+c=x2-3x+1
2bx=-3x
b=-3/2
b2+c=1
c=1-9/2
c=-7/2
b+c=-3/2+-7/2=-10/2=-5
What is the value of c if x(3x+1)<c only when x is (-7/3, 2)
3x2+x<c
3x2+x-c<0
3=a, 1=b, -c=c, x=(-b+\sqrt(b2-4ac))/2a
x<(-1+\sqrt(1+12c))/6
Test: x=-7/3
-14<-1+\sqrt(1+12c)
-13<+\sqrt1+12c
169>1+12c
168>12c
c<14
Test: x=2
12<-1+\sqrt(1+12c)
13<+\sqrt(1+12c)
169<1+12c
168<12c
c>14
If c<14 and c>14, then c=14
+129840 What is the value of c if x * (3x + 1) < c and x lies on the interval (-7/3, 2)
This is a parabola that turns upward
At x = -7/3, x (3x + 1 ) = 14
At x = 2, x (3x + 1) = 14
So.....on the interval (-7/3, 2), because of parabolic symmetry, all f(x) will be < 14....thus... the inequality will be true whenever c = 14
